General decay of nonlinear viscoelastic Kirchhoff equation with Balakrishnan-Taylor damping and logarithmic nonlinearity. (English) Zbl 1428.35037
Summary: This work deals with the study of a new class of nonlinear viscoelastic Kirchhoff equation with Balakrishnan-Taylor damping and logarithmic nonlinearity. A decay result of the energy of solutions for the problem without imposing the usual relation between a certain relaxation function and its derivative is established. This result generalizes earlier ones to an arbitrary rate of decay, which is not necessarily of exponential or polynomial decay.
MSC:
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35L72 | Second-order quasilinear hyperbolic equations |
| 35R09 | Integro-partial differential equations |
References:
| [1] | CavalcantiM, CavalcantiVND, FilhoJSP, SorianoJA. Exponential decay for the solution of semi linear viscoelastic wave equations with localized damping. Electronic J Differ Equ. 2002;2002(44):1‐14. · Zbl 0997.35037 |
| [2] | CavalcantiM, CavalcantiVND, MartinezP. General decay rate estimates for viscoelastic dissipative systems. Nonlinear Anal. 2008;68:177‐193. · Zbl 1124.74009 |
| [3] | KirchhoffG. Vorlesungen uber Mechanik. Leipzig: Tauber; 1883. |
| [4] | MezouarN, BoulaarasS. Global existence of solutions to a viscoelastic non‐degenerate Kirchhoff equation. Appl Anal. 2018a. https://doi.org/10.1080/00036811.2018.1544621 · Zbl 1458.35255 · doi:10.1080/00036811.2018.1544621 |
| [5] | MezouarN, BoulaarasS. Global existence and decay of solutions for a class of siscoelastic Kirchhoff equation. Bull Malays Math Sci Soc. 2018b. https://doi.org/10.1007/s40840-018-00708-2 · Zbl 1435.35261 · doi:10.1007/s40840-018-00708-2 |
| [6] | MuC, MaJ. On a system of nonlinear wave equations with Balakrishnan‐Taylor damping. Z Angew Math Phys. 2014;65:91‐113. · Zbl 1295.35309 |
| [7] | MesloubF, BoulaarasS. General decay for a viscoelastic problem with not necessarily decreasing kernel. J Appl Math Comput. 2018;58:647‐665. · Zbl 1403.35050 |
| [8] | OuchenaneD, BoulaarasS, MesloubF. General decay for a viscoelastic problem with not necessarily decreasing kernel. Appl Anal. 2018. https://doi.org/10.1080/00036811.2018 · doi:10.1080/00036811.2018 |
| [9] | MedjdenM, TatarN‐e. Asymptotic behavior for a viscoelastic problem with not necessarily decreasing kernel. Appl Math Comput. 2005;167:1221‐1235. · Zbl 1076.74022 |
| [10] | BoumazaN, BoulaarasS. General decay for Kirchhoff type in viscoelasticity with not necessarily decreasing kernel. Math Methods Appl Sci. 2018;41:6050‐6069. · Zbl 1415.35038 |
| [11] | ZaraiA, TatarN‐e. Global existence and polynomial decay for a problem with Balakrishnan‐Taylor damping. Arch Math (BRNO). 2010;46:157‐176. · Zbl 1240.35330 |
| [12] | NakaoM. Decay of solutions of some nonlinear evolution equation. J Math Anal Appl. 1977;60:542‐549. · Zbl 0376.34051 |
| [13] | OnoK. Global existence, decay, and blow‐up of solutions for some mildly degenerate nonlinear Kirchhoff strings. J Differ Equ. 1997;137:273‐301. · Zbl 0879.35110 |
| [14] | LiMR, TsaiLY. Existence and nonexistence of global solutions of some systems of semilinear wave equations. Nonlinear Anal. 2003;54:1397‐1415. · Zbl 1026.35067 |
| [15] | CavalcantiM, FilhoVND, FilhoJSP, SorianoJA. Existence and uniform decay rates for viscoelastic problems with nonlinear boundary damping. Differential Integral Equ. 2001;14:85‐116. · Zbl 1161.35437 |
| [16] | BalakrishnanAV, TaylorLW. Distributed parameter nonlinear damping models for flight structures. In: Proceedings “Damping 89”, Flight Dynamics Lab and Air Force Wright Aeronautical Labs, Washington: WPAFB; 1989. |
| [17] | BassRW, ZesD. Spillover nonlinearity, and flexible structures. The Fourth NASA Workshop on Computational Control of Flexible Aerospace Systems. Washington: NASA Conference Publication 10065; 1991:1‐14. |
| [18] | TatarN‐e, ZaraiA. Exponential stability and blow up for a problem with Balakrishnan‐Taylor damping. Demonstr Math, XLIV. 2011;1:67‐90. · Zbl 1227.35074 |
| [19] | CavalcantiM, OquendoHP. Frictional versus viscoelastic damping in a semi linear wave equation. SIAM J Control Optim. 2003;42:1310‐132. https://doi.org/10.1137/S0363012902408010 · Zbl 1053.35101 · doi:10.1137/S0363012902408010 |
| [20] | BerrimiS, MessaoudiSA. Exponential decay for the solution of semilinear viscoelastic wave equations with localized damping. Electronic J Differ Equ. 2004;2004(88):1‐10. · Zbl 1055.35020 |
| [21] | LiuW. Exponential or polynomial decay of solutions to a viscoelastic equation with nonlinear localized damping. J Appl Math Comput. 2010;32:59‐68. https://doi.org/10.1007/s12190-009-0232-y · Zbl 1194.35046 · doi:10.1007/s12190-009-0232-y |
| [22] | ZaraiA, DraifiaA, BoulaarasS. Blow up of solutions for a system of nonlocal singular viscoelastic equations. Appl Anal. 2018;97:2231‐2245. · Zbl 1403.35060 |
| [23] | Al‐GharabliMM. A general decay result of a viscoelastic equation with infinite history and nonlinear damping. Appl Anal. 2018;97:382‐399. · Zbl 1460.74010 |
| [24] | BoulaarasS, DraifiaA, AlneggaM. Polynomial decay rate for Kirchhoff type in viscoelasticity with logarithmic nonlinearity and not necessarily decreasing kernel. Symmetry. 2019;11(2):226. https://doi.org/10.3390/sym11020226 · Zbl 1416.35156 · doi:10.3390/sym11020226 |
| [25] | KafiniM. Uniform decay of solutions to Cauchy viscoelastic problems with density. Elecron J Differ Equ. 2011;2011(93):1‐9. · Zbl 1227.35065 |
| [26] | SunF, WangM. Existence and nonexistence of global solutions for a nonlinear hyperbolic system with damping. Nonlinear Anal. 2007;66:2889‐2910. · Zbl 1129.35046 |
| [27] | Alabau‐BoussouiraF, CannarsaP. A general method for proving sharp energy decay rates for memory‐dissipative evolution equations. C R Math Acad Sci Paris Ser. 2009;I(347):867‐872. · Zbl 1179.35058 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
